Introduction to Variational Inequalities and Their Applications (eBook)

Front Cover 1
An Introduction to Variational Inequalities and Their Applications 4
Copyright Page 5
Contents 8
Preface 12
Glossary of Notations 14
Introduction 16
Chapter I. Variational Inequalities in Rn 22
1. Fixed Points 22
2. The Characterization of the Projection onto a Convex Set 23
3. A First Theorem about Variational Inequalities 26
4. Variational Inequalities 28
5. Some Problem Which Lead to Variational Inequalities 30
Comments and Bibliographical Notes 33
Exercises 33
Chapter II. Variational Inequalities in Hilbert Space 38
1. Bilinear Forms 38
2. Existence of a Solution 39
3. Truncation 42
4. Sobolev Spaces and Boundary Value Problems 43
5. The Weak Maximum Principle 50
6. The Obstacle Problem: First Properties 55
7. The Obstacle Problem in the One Dimensional Case 62
Appendix A. Sobolev Spaces 64
Appendix B. Solutions to Equations with Bounded Measurable Coefficients 77
Appendix C. Local Estimates of Solutions 81
Appendix D. Hölder Continuity of the Solutions 87
Comments and Bibliographical Notes 91
Exercises 92
Chapter III. Variational Inequalities for Monotone Operators 98
1. An Abstract Existence Theorem 98
2. Noncoercive Operators 102
3. Semilinear Equations 108
4. Quasi-Linear Operators 109
Comments and Bibliographical Notes 115
Exercises 116
Chapter IV. Problems of Regularity 120
1. Penalization 120
2. Dirichlet Integral 121
3. Coercive Vector Fields 128
4. Locally Coercive Vector Fields 131
5. Another Penalization 135
6. Limitation of Second Derivatives 139
7. Bounded Variation of Au 145
8. Lipschitz Obstacles 149
9. A Variational Inequality with Mixed Boundary Conditions 154
Appendix A. Proof of Theorem 3.3 158
Comments and Bibliographical Notes 161
Exercises 162
Chapter V. Free Boundary Problems and the Coincidence Set of the Solution 164
I. Introduction 164
2. The Hodograph and Legendre Transformations 168
3. The Free Boundary in Two Dimensions 170
4. A Remark about Singularities 181
5. The Obstacle Problem for a Minimal Surface 182
6. The Topology of the Coincidence Set When the Obstacle Is Concave 188
7. A Remark about the Coincidence Set in Higher Dimensions 193
Comments and Bibliographical Notes 196
Exercises 197
Chapter VI. Free Boundary Problems Governed by Elliptic Equations and Systems 199
I. Introduction 199
2. Hodograph and Legendre Transforms: The Theory of a Single Equation 200
3. Elliptic Systems 205
4. A Reflection Problem 217
5. Elliptic Equations Sharing Cauchy Data 219
6. A Problem of Two Membranes 227
Comments and Bibliographical Notes 233
Exercises 233
Chapter VII. Applications of Variational Inequalities 237
1. Introduction 237
2. A Problem in the Theory of Lubrication 238
3. The Filtration of a Liquid through a Porous Medium 242
4. The Resolution of the Filtration Problem by Variational Inequalities 250
5. The Filtration of a Liquid through a Porous Medium with Variable Cross Section 257
6. The Resolution of the Filtration Problem in Three Dimensions 264
7. Flow past a Given Profile: The Problem in the Physical Plane 272
8. Flow past a Given Profile: Resolution by Variational Inequalities 275
9. The Deflection of a Simply Supported Beam 285
Comments and Bibliographical Notes 288
Exercises 289
Chapter VIII. A One Phase Stefan Problem 293
1. Introduction 293
2. Existence and Uniqueness of the Solution 296
3. Smoothness Properties of the Solution 304
4. The Legendre Transform 312
Comments and Bibliographical Notes 314
Bibliography 315
Index 324